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On positive real numbers, (a*b=a^{\log b}), where (\log) has base (10). When is this operation commutative?धनात्मक वास्तविक संख्याओं पर (a*b=a^{\log b}) दिया है, जहाँ (\log) का आधार (10) है। यह संक्रिया क्रमविनिमेय कब होगी?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: For all (a,b>0)सभी (a,b>0) के लिए
Step 1: Commutativity needs (a*b=b*a). Step 2: Here (a^{\log b}=b^{\log a}). Taking (\log) on both sides gives ((\log a)(\log b)=(\log b)(\log a)), always true. Step 3: For exponential operations, logarithms often simplify comparison.चरण 1: क्रमविनिमेयता के लिए (a*b=b*a) चाहिए। चरण 2: (a^{\log b}=b^{\log a})। दोनों ओर (\log) लेने पर ((\log a)(\log b)=(\log b)(\log a)), जो सदा सत्य है। चरण 3: घात वाले प्रश्नों में लघुगणक लेने से तुलना सरल हो जाती है।
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